scholarly journals Partial duality and Bollobás and Riordan’s ribbon graph polynomial

2010 ◽  
Vol 310 (1) ◽  
pp. 174-183 ◽  
Author(s):  
Iain Moffatt
Keyword(s):  
Ribbon Graph ◽  
Graph Polynomial

A Graph Polynomial Approach to Primitivity

2013 ◽  
pp. 153-164 ◽  
Author(s):  
Francine Blanchet-Sadri ◽  
Michelle Bodnar ◽  
Nathan Fox ◽  
Joe Hidakatsu
Keyword(s):  
Graph Polynomial

scholarly journals A Graph Polynomial for Independent Sets of Bipartite Graphs

2012 ◽  
Vol 21 (5) ◽  
pp. 695-714 ◽  
Author(s):  
Q. GE ◽  
D. ŠTEFANKOVIČ
Keyword(s):  
Riemann Hypothesis ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Rational Points ◽  
Perfect Matchings ◽  
Generalized Riemann Hypothesis ◽  
Graph Polynomial ◽  
Exact Evaluation

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.

scholarly journals The Rotor-Routing Torsor and the Bernardi Torsor Disagree for Every Non-Planar Ribbon Graph

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Changxin Ding
Keyword(s):  
Spanning Trees ◽  
Picard Group ◽  
Ribbon Graph

Let $G$ be a ribbon graph. Matthew Baker and Yao Wang proved that the rotor-routing torsor and the Bernardi torsor for $G$, which are two torsor structures on the set of spanning trees for the Picard group of $G$, coincide when $G$ is planar. We prove the conjecture raised by them that the two torsors disagree when $G$ is non-planar. 

scholarly journals Ribbon graphs and bialgebra of Lagrangian subspaces

2016 ◽  
Vol 25 (12) ◽  
pp. 1642006 ◽  
Author(s):  
Victor Kleptsyn ◽  
Evgeny Smirnov
Keyword(s):  
Vector Space ◽  
Symplectic Form ◽  
Chord Diagram ◽  
Ribbon Graph ◽  
Dimensional Vector Space ◽  
Ribbon Graphs ◽  
Chord Diagrams ◽  
Intersection Matrix ◽  
Lagrangian Subspace ◽  
Lagrangian Subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

scholarly journals KHOVANOV–ROZANSKY GRAPH HOMOLOGY AND COMPOSITION PRODUCT

2008 ◽  
Vol 17 (12) ◽  
pp. 1549-1559 ◽  
Author(s):  
E. WAGNER
Keyword(s):  
Recursive Formula ◽  
Graph Polynomial ◽  
Graph Homology ◽  
Composition Product

In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov–Rozansky graph homology.

scholarly journals General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 149-155
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Complete Graphs ◽  
Dominating Sets ◽  
Dihedral Groups ◽  
Graph Polynomial ◽  
Central Elements ◽  
Domination Polynomial ◽  
Class Graph

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

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